The scaling equation of state of an Ising-like ferromagnet is derived by an expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$, where
d is the dimension of space. The result is compared with numerical calculations on the three-dimensional Ising model. It is also established that the "linear model" is exact up to order ${\ensuremath{\epsilon}}^{2}$.
In recent years there have been many measurements of the scaling-law equation of state for different materials, and the "scaling function" so obtained has generally been fit by an empirical equation involving the selection of several adjustable parameters. We propose a method for calculating, directly from high-temperature series expansions, the function
h(x) that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously,
h(x) has been calculated only for the
S=21 Ising model, but the method is not generalizable to the case of the Heisenberg model because it relies upon the use of low-temperature expansions as well, and these are not known for the Heisenberg model. First we calculate
h(x) for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model
h(x) agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate
h(x) in its entire region of definition for the
S=21 Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where
S denotes the spin quantum number. The accuracy of our resulting expressions is limited by the finite number of known terms in the corresponding high-temperature series expansions, but it is generally of the order of a few percent. In Paper II the scaling functions calculated here are compared with experiment and with the predictions of the universality hypothesis.
The critical exponents $\ensuremath{\gamma}$, $\ensuremath{\eta}$ and the "crossover index" $\ensuremath{\phi}$ are computed for generalized classical Heisenberg models with
n internal degrees of freedom as an exact expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ (
d is the number of space dimensions). Results are obtained to order ${\ensuremath{\epsilon}}^{2}$ for $\ensuremath{\gamma}$ and to order ${\ensuremath{\epsilon}}^{3}$ for $\ensuremath{\eta}$. The results to this order for the three-dimensional Ising case ($n=\ensuremath{\epsilon}=1$) are $\ensuremath{\gamma}=1.244$ and $\ensuremath{\eta}=0.037$.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
Making use of the scaling properties of $H(\ensuremath{\epsilon}, M)={M}^{\ensuremath{\delta}}h(\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}})$, it is shown that the "scaling function"
h(x) can be expressed rigorously as $\frac{h(x)}{h(0)}={[\frac{({x}_{0}+x)}{{x}_{0}}]}^{\ensuremath{\beta}(\ensuremath{\delta}\ensuremath{-}1)}$ in the neighborhood of the critical point. This result is in agreement with the known mean-field-model prediction, directly obtainable from power-series expansion, and with accurate numerical results for the Ising and Heisenberg models.
A method is presented for the convergent evaluation of the Feynman diagrams for the free energy of an Ising-type system. The method involves three first-order Wilson-type differential equations whose solution near and at the fixed point yield the critical parameters
y and $\ensuremath{\eta}$. The method is discussed only for a system with no long-range order. That is, we consider only the case
H=0,
T>Tc. The $\ensuremath{\epsilon}$ expansions for
y and $\ensuremath{\eta}$ are evaluated to second order, and
y and $\ensuremath{\eta}$ are found exactly for the spherical model.